170 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY 



H 2 : R 2 : : x* : r 2 ; 

 or from this, by equating and dividing, we get 



r *-^. 



~ H 2 ' 



consequently, by the principles of mensuration, the area of the section, 

 on which the pressure is proposed to be a maximum, becomes 



3.1416R 2 * 2 

 -rf -- ; 



therefore, the pressure upon its surface is 

 3.1416R a * 2 s 



Now, because the whole of the quantities which enter this equa- 

 tion are constant, excepting x, and the bracketted expression 

 {Rcot.^-f H x} which is affected by it; it follows, that the value 

 of p varies as rc 2 {RCOt.0-|- H x}, and consequently, is a maximum, 

 when the quantity which limits its variation is a maximum ; hence 

 we have 



a? 2 {R cot.^ -|- H x} zz a maximum. 



Let the above expression for the maximum be thrown into fluxions, 

 and we shall obtain 



2 (R cot.0 -f- H) xx 3a? a i =. ; 



therefore, by transposing and expunging the common quantities, we 

 get 



3a?i=2(RCot.0+ H), 

 and finally, by division, we obtain 



a? = |-(RCOt.^ + H). (132). 



169. The practical rule for reducing this equation, may be expressed 

 in words at length in the following manner. 



RULE. Multiply the natural cotangent of the angle which 

 the axis of the cone makes with the horizon, by the radius of 

 the vessel's base, and to the product add the altitude or axis 

 of the cone ; then, two thirds of the sum will give the distance 

 of the section, on which the pressure is a maximum, from the 

 vertex of the cone. 



170. EXAMPLE. A conical vessel whose altitude is 20 inches, and 

 the radius of its base 8 inches, is filled with fluid and so inclined, that 

 its axis makes with the horizontal line passing through the extremity 

 of the diameter of its base, an angle of 48 degrees ; on what section, 

 parallel to the base is the pressure a maximum ? 



